I've figured out how to use my new remote shutter release with my new Panasonic GH3 when using a fixed shutter setting, but I don't know how to use it so that the shutter stays open until I take my finger off the remote button. I tried setting the camera to manual first, but that didn't work.

There is a shutter speed called Bulb Mode that will keep the shutter open while the shutter button is depressed. There is a limit to how long the shutter will be allowed to remain open. It appears that the GH3 at least extends the 120-second bulb-mode time limitation of the GH2: Panasonic's specifications page lists this limit at around 60 minutes.